Solve for $x$ : $ 8|x - 6| + 4 = 3|x - 6| + 6 $
Solution: Subtract $ {3|x - 6|} $ from both sides: $ \begin{eqnarray} 8|x - 6| + 4 &=& 3|x - 6| + 6 \\ \\ { - 3|x - 6|} && { - 3|x - 6|} \\ \\ 5|x - 6| + 4 &=& 6 \end{eqnarray} $ Subtract ${4}$ from both sides: $ \begin{eqnarray} 5|x - 6| + 4 &=& 6 \\ \\ { - 4} &=& { - 4} \\ \\ 5|x - 6| &=& 2 \end{eqnarray} $ Divide both sides by ${5}$ $ \dfrac{5|x - 6|} {{5}} = \dfrac{2} {{5}} $ Simplify: $ |x - 6| = \dfrac{2}{5}$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x - 6 = -\dfrac{2}{5} $ or $ x - 6 = \dfrac{2}{5} $ Solve for the solution where $x - 6$ is negative: $ x - 6 = -\dfrac{2}{5} $ Add ${6}$ to both sides: $ \begin{eqnarray} x - 6 &=& -\dfrac{2}{5} \\ \\ {+ 6} && {+ 6} \\ \\ x &=& -\dfrac{2}{5} + 6 \end{eqnarray} $ Change the ${ + 6}$ to an equivalent fraction with a denominator of $5$ $ x = - \dfrac{2}{5} {+ \dfrac{30}{5}} $ $ x = \dfrac{28}{5} $ Then calculate the solution where $x - 6$ is positive: $ x - 6 = \dfrac{2}{5} $ Add ${6}$ to both sides: $ \begin{eqnarray} x - 6 &=& \dfrac{2}{5} \\ \\ {+ 6} && {+ 6} \\ \\ x &=& \dfrac{2}{5} + 6 \end{eqnarray} $ Change the ${ + 6}$ to an equivalent fraction with a denominator of $5$ $ x = \dfrac{2}{5} {+ \dfrac{30}{5}} $ $ x = \dfrac{32}{5} $ Thus, the correct answer is $x = \dfrac{28}{5} $ or $x = \dfrac{32}{5} $.